Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#12881 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | #432 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(4,5)(7,8,11,18,12,15,27,31,13,16,26,32,10,17,24,33)(9,19,21,34,22,29,36,38,23,14,25,30,20,28,35,37);(3,4,6,5)(7,9,10,20,13,23,12,22)(8,14,17,29,16,19,15,28)(11,25,24,36,26,21,27,35)(18,34,33,37,32,30,31,38);(3,5,6,4)(8,15,16,17)(11,26)(14,28,19,29)(18,33,32,31)(21,25)(24,27)(30,38,34,37)(35,36)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401262);